# Full text of "Treatise On Analysis Vol-Ii"

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```6    POSITIVE LINEAR FORMS, POSITIVE HILBERT FORMS       349
(ii) For all x, y in A we have

(15.6.2.2)                        \f(y*x)\2 £/(***)/(/».
(iii) If A has a unit element e, then

(15.6.2.3)                             /(**) =7(7),

(15.6.2.4)                            |/(x)|2 £/(«)/(***).

To prove (i), express that

g(x + y9x+y) = g(x, x) + g(x9 y) + #0>, x) + g(y, y)
is real: by virtue of the hypothesis (15.6.1), we obtain

changing x into ix, this becomes 3\$(g(x9 y)) = @%(g(y, x)), hence g(y, x) =
g(x, y). Assertion (ii) is the Cauchy-Schwarz inequality (6.2.1 ) applied to g.
The relations (iii) are obtained by replacing y by e in (15.6.2.1) and (15.6.2.2),
and using (15.4.2).

The hermitian form g so obtained from / is not arbitrary, because it
clearly satisfies the relation

(15.6.3)                               g(xy,z)=g(y,x*z)

for all x, y, z in A. We therefore make the following definition:

A positive Hilbert form on the algebra with involution A is a positive

hermitian form on A x A which satisfies the relation (15.6.3).

If A has a unit element, every positive Hilbert form g comes as above from

a positive linear form/: we have only to define /(jc) = g(x, e). But we shall see

later that this is no longer the case when A has no unit element (15.7.4).

We shall now show that there are remarkable relationships between
Hilbert forms on A x A and representations of A.

In the first place, every representation s^ U(s) of A in a Hilbert space H
gives rise to positive linear forms (and hence to positive Hilbert forms) in the
following way: for each XQ e H, if we put

(15.6.4)                               /*(*)= («(s) • x0 | *0)f(y*x) is a positive hermitian form on
```